X-ray Binaries

An X-ray binary is a star system caught in the act of feeding a compact object. One member is a white dwarf, neutron star, or stellar-mass black hole; the other is an ordinary star that is losing gas — through a wind, or by swelling until its outer layers spill across to the compact companion. The transferred gas cannot fall straight in, because it carries the angular momentum of the orbit. It settles instead into a flattened accretion disc and spirals inward slowly, giving up angular momentum to viscous stresses and giving up gravitational energy to heat. By the time the gas reaches the bottom of the potential well — the surface of a neutron star, or the innermost stable orbit around a black hole — it has been heated to $10^7$ – $10^8\,\mathrm{K}$ , and it radiates that heat as X-rays. The energy budget is what makes these systems extreme. Because the accretor is tiny, its potential well is extraordinarily deep: a gram of gas dropped onto a neutron star releases about a fifth of its rest-mass energy, roughly twenty times what hydrogen fusion yields, so a modest trickle of matter lights up a point source at $10^{36}$ – $10^{38}\,\mathrm{erg\,s^{-1}}$ . And there is a natural ceiling: when the X-rays' outward push on the plasma balances gravity, the source reaches its Eddington limit, set only by the accretor's mass. This article develops the two governing scales — the accretion luminosity $GM\dot M/R$ and the Eddington limit — and then shows how X-ray binaries became the most precise weighing scales in astrophysics: the instruments that measure neutron-star masses, confirm that stellar-mass black holes exist, and record the recycling of old neutron stars into millisecond pulsars.

Key takeaways

An X-ray binary is a compact object — white dwarf, neutron star, or black hole — accreting gas from a stellar companion and radiating the released gravitational energy as X-rays from a hot inner accretion flow.
The engine is accretion power, $L_{\mathrm{acc}}=GM\dot M/R$ ; for a neutron star this is $\sim0.2\,\dot M c^2$ , about twenty times more energy per gram than hydrogen fusion — which is why so little fuel makes so bright a source.
Systems split into low-mass X-ray binaries (LMXBs), fed by Roche-lobe overflow through the inner Lagrangian point $L_1$ , and high-mass X-ray binaries (HMXBs), fed by the wind of a massive OB companion.
The Eddington luminosity $L_{\mathrm{Edd}}\simeq1.3\times10^{38}(M/M_\odot)\,\mathrm{erg\,s^{-1}}$ caps steady spherical accretion, and the disc radiates a multi-temperature blackbody with $T_{\mathrm{eff}}\propto R^{-3/4}$ .
Radial-velocity data give the mass function $f(M)=P_{\mathrm{orb}}K_c^3/2\pi G\le M_X$ , an inclination-free lower bound on the accretor mass that identifies a black hole when it exceeds the neutron-star maximum.
Key idea: the engine of an X-ray binary is gravitational — gas falling to a compact object of mass $M$ and radius $R$ releases $\sim GM/R$ per unit mass, so $L_{\mathrm{acc}}=GM\dot M/R$ is enormous precisely because $R$ is small, and it is throttled from above by the Eddington limit where radiation pressure balances gravity.

The core insight

The engine of an X-ray binary is gravitational: gas falling from rest at infinity onto a compact object of mass $M$ and radius $R$ releases $\sim GM/R$ per unit mass, so the luminosity $L_{\mathrm{acc}}=GM\dot M/R$ is huge precisely because $R$ is small — and it is throttled from above by the Eddington limit, where the outward push of the radiation balances the inward pull of gravity.

Where this lives today

X-ray binaries remain among the most productive laboratories in astrophysics. They furnish the sharpest neutron-star masses: pulsar timing and eclipse modelling reach $\sim2\,M_\odot$ , and each high value chops away at candidate dense-matter equations of state through the Tolman–Oppenheimer–Volkoff maximum — a programme sharpened since GW170817 added an independent tidal-deformability handle. Black-hole XRBs anchor accretion-state theory: during outbursts a source loops through the hardness–intensity diagram between a soft, disc-dominated state near $1\,\mathrm{keV}$ and a hard, corona-dominated state that launches a compact radio jet, and this disc–jet coupling scales up by $10^{8}$ in mass to active galactic nuclei. Relativistically broadened iron $\mathrm{K}\alpha$ lines and quasi-periodic oscillations, now probed by NICER and IXPE, measure black-hole spins and map the strong-field flow within a few gravitational radii. LMXBs are the birthplace of millisecond pulsars; transitional millisecond pulsars, caught swinging between accretion- and rotation-powered states over months, have clinched the recycling scenario in real time. And ultraluminous X-ray sources, some known to be neutron stars radiating hundreds of times their Eddington limit, keep the question of how nature evades the cap firmly open.

The physical picture

Picture dropping a marble down a very deep, very narrow well. On the Earth's surface the well is shallow and the marble arrives gently; on a neutron star it gives back about a fifth of the marble's rest-mass energy as heat before the bottom. That efficiency — not any nuclear reaction — powers an X-ray binary. But the gas cannot dive straight in, because it is in orbit and carries angular momentum. It sheds that slowly, in a flattened accretion disc where friction between neighbouring rings bleeds orbital energy into heat. The innermost rings sit deepest and are hottest, so their glow lands in the X-ray band while the cool outer disc shines only in the optical and ultraviolet. The first figure is the anatomy of a low-mass system: a donor swollen to fill its Roche lobe, a gas stream leaking through the inner Lagrangian point, and a disc carrying it down to the compact object.

Figure

Anatomy of a low-mass X-ray binary: the donor fills its Roche lobe and spills gas through the inner Lagrangian point L₁; the stream feeds an accretion disc that spirals down to the compact object, where the released binding energy L = GMṀ/R emerges as X-rays from gas at 10⁷–10⁸ K

There is a self-limiting feedback built into the same X-rays. The outward radiation scatters off the plasma's electrons and pushes them back, and since electrostatic coupling drags the protons along, the whole flow feels an outward force; raise the accretion rate and the luminosity climbs until, at the Eddington limit, that force cancels gravity and steady inflow is choked off. The accretor's identity shows in the details: a neutron star has a hard surface where matter can detonate as a thermonuclear (Type-I) burst and a magnetic field that funnels the flow onto its poles to pulse; a black hole has neither, only a horizon that swallows the flow silently. The second figure is the disc's thermometer — how its temperature runs with radius, and why only the inner edge glows in X-rays.

Figure

The thin-disc temperature runs as T ∝ R^−3⁄4, so over the four decades of radius from a 10 km inner edge to a 10⁵ km outer edge the disc cools from ~10⁷ K to ~10⁴ K: only the innermost annuli, at kT ~ 1 keV, radiate in the X-ray band, while the cool outer disc glows in the optical and ultraviolet

The core equations

The engine is a single balance: the rate at which gravitational binding energy is liberated equals the emitted luminosity. Equation (1) multiplies the specific binding energy at the accretor's surface by the mass supply rate.

L_{\mathrm{acc}}=\dfrac{GM\dot M}{R} \qquad (1)

where: $L_{\mathrm{acc}}$ — luminosity from released binding energy; $M$ — compact-accretor mass; $R$ — radius where the energy is liberated (the surface, or the innermost stable orbit for a black hole); $\dot M$ — mass accretion rate; $G$ — Newton's constant. Reading: $L$ is large not because $\dot M$ is large — it need not be — but because $R$ is tiny, making $GM/R$ enormous. Rewriting (1) in units of the rest-mass energy supplied, $\dot M c^2$ , isolates a dimensionless efficiency $\eta$ that compares accretors against each other and against nuclear burning.

L=\eta\,\dot M c^2,\qquad \eta=\dfrac{GM}{Rc^2} \qquad (2)

where: $\eta$ — radiative efficiency, the fraction of rest-mass energy released; $c$ — speed of light. Reading: a neutron star has $\eta\approx0.2$ ; a Schwarzschild black hole radiates $\eta\approx0.057$ at its innermost stable orbit, a maximal Kerr hole up to $\approx0.42$ — all dwarfing the $0.007$ of fusion. The luminosity cannot climb without limit. Setting the outward radiation force on ionized hydrogen equal to gravity gives a radius-independent, mass-only ceiling — the Eddington luminosity.

L_{\mathrm{Edd}}=\dfrac{4\pi G M m_p c}{\sigma_T}\simeq1.26\times10^{38}\left(\dfrac{M}{M_\odot}\right)\,\mathrm{erg\,s^{-1}} \qquad (3)

where: $m_p$ — proton mass (the mass per free electron in ionized hydrogen); $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ — Thomson cross-section; $L_{\mathrm{Edd}}$ — maximum luminosity for steady spherical accretion. Reading: because $L_{\mathrm{Edd}}\propto M$ , a source known to radiate at its limit is a distance-independent scale for the accretor mass. Dividing the Eddington luminosity by the efficiency converts it into the maximum steady supply rate — the Eddington accretion rate, the fuel-delivery ceiling.

\dot M_{\mathrm{Edd}}=\dfrac{L_{\mathrm{Edd}}}{\eta c^2}=\dfrac{4\pi G M m_p}{\eta\,\sigma_T c} \qquad (4)

where: $\dot M_{\mathrm{Edd}}$ — accretion rate that produces $L_{\mathrm{Edd}}$ . Reading: for a canonical neutron star ( $\eta\approx0.2$ ) this is $\sim2\times10^{-8}\,M_\odot\,\mathrm{yr^{-1}}$ , and bright persistent LMXBs sit within an order of magnitude of it. In a steady, thin disc the viscous dissipation at each radius, set by angular-momentum transport, is re-radiated as a local blackbody from both faces. Solving that balance gives the temperature profile.

T_{\mathrm{eff}}(R)=\left[\dfrac{3GM\dot M}{8\pi\sigma R^3}\left(1-\sqrt{\tfrac{R_\ast}{R}}\right)\right]^{1/4}\propto R^{-3/4} \qquad (5)

where: $T_{\mathrm{eff}}(R)$ — local effective temperature at radius $R$ ; $R_\ast$ — inner disc edge (surface or innermost stable orbit); $\sigma$ — Stefan–Boltzmann constant. Reading: far from the edge the correction fades and $T\propto R^{-3/4}$ , so the spectrum is a superposition of blackbodies — a multi-colour disc whose hot inner annuli make the X-rays. To weigh the accretor we turn from energetics to dynamics. When only the companion's lines are visible, Kepler's laws combine its period and radial-velocity amplitude into the mass function, which bounds the unseen mass from below.

f(M)=\dfrac{M_X^3\sin^3 i}{(M_X+M_c)^2}=\dfrac{P_{\mathrm{orb}}\,K_c^3}{2\pi G} \qquad (6)

where: $M_X$ — compact-object mass; $M_c$ — donor mass; $i$ — orbital inclination; $P_{\mathrm{orb}}$ — orbital period; $K_c$ — radial-velocity semi-amplitude of the donor. Reading: since $\sin i\le1$ and $M_c>0$ , the measurable right-hand side is an absolute lower bound, $M_X\ge f(M)$ — and $f(M)>3\,M_\odot$ forces a black hole. Mass transfer by overflow begins when the donor swells to fill its critical equipotential — its Roche lobe. Eggleton's fit gives that lobe's effective radius across all mass ratios.

R_L\simeq\dfrac{0.49\,q^{2/3}}{0.6\,q^{2/3}+\ln\!\left(1+q^{1/3}\right)}\,a,\qquad q=\dfrac{M_c}{M_X} \qquad (7)

where: $R_L$ — effective Roche-lobe radius; $a$ — orbital separation; $q$ — mass ratio, donor to accretor. Reading: when the donor's radius reaches $R_L$ , gas spills through $L_1$ onto the accretor — the mechanism defining a low-mass X-ray binary. For a magnetized neutron star the disc does not even reach the surface: the magnetic pressure grows inward as $r^{-6}$ and overwhelms the ram pressure of the flow at the magnetospheric (Alfvén) radius $r_m=(\mu^4/2GM\dot M^2)^{1/7}$ , inside which the field channels matter onto the poles to produce X-ray pulsations — derived in full below.

Derivation AThe Eddington luminosity

The goal is to find the luminosity at which a source's own radiation halts the accretion that powers it — where the outward radiative force on the infalling plasma exactly cancels gravity. We use a fully ionized hydrogen plasma, in which radiation couples to the light electrons while gravity acts on the heavy protons and electrostatics locks the two into one fluid; because both forces fall as the inverse square of radius, their ratio is a pure, radius-independent number — a ceiling on steady spherical accretion set only by the accretor's mass.

Step 1. For a source of luminosity $L$ , an electron at radius $r$ intercepts the radiation momentum flux $L/(4\pi r^2 c)$ over the Thomson cross-section, so the outward force is $F_{\mathrm{rad}}=\sigma_T L/(4\pi r^2 c)$ . Why: Thomson scattering off the light electrons, not the protons, couples the radiation to the plasma — the cross-section scales as the inverse square of particle mass.

Step 2. Gravity pulls chiefly on the protons: for one electron–proton pair, $F_{\mathrm{grav}}=GM(m_p+m_e)/r^2\approx GMm_p/r^2$ . The two cannot separate — any charge separation sets up a restoring electric field — so they act as one fluid element. Why: this electrostatic locking is the crux; without it the electrons would simply be blown away and the argument would collapse.

Step 3. Set the forces equal, $\sigma_T L/(4\pi r^2 c)=GMm_p/r^2$ ; both sides carry the same $r^{-2}$ , which cancels exactly. Why: the balance holds at every radius at once, so exceeding it chokes the flow everywhere — a genuine luminosity limit, not a statement about one shell.

Step 4. Solve for the limit: $L_{\mathrm{Edd}}=4\pi GMm_pc/\sigma_T$ . Inserting $M_\odot=1.99\times10^{33}\,\mathrm{g}$ , $m_p=1.67\times10^{-24}\,\mathrm{g}$ , $c=3\times10^{10}\,\mathrm{cm\,s^{-1}}$ , $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ gives $1.26\times10^{38}\,\mathrm{erg\,s^{-1}}$ per solar mass. Why: the coefficient turns an abstract balance into an observational yardstick — $10^{38}\,\mathrm{erg\,s^{-1}}$ points to a solar-mass accretor. *Verified: $4\pi GM_\odot m_pc/\sigma_T=1.26\times10^{38}\,\mathrm{erg\,s^{-1}}$ , matching Frank, King & Raine and Carroll & Ostlie.*

Go deeper · Derivation B · 1 minThe compact-object mass from the mass function

Derivation BThe compact-object mass from the mass function

The goal is to show that radial-velocity spectroscopy of the visible star alone — with no image of the compact object and no knowledge of the inclination — places a firm lower bound on the accretor mass, often enough to prove it is a black hole. We measure only two numbers, the orbital period and the donor's velocity amplitude, and combine them with Kepler's third law and the centre-of-mass condition to isolate the mass function — a pure lower limit on the hidden mass regardless of the two unknowns, inclination and donor mass.

Step 1. The donor's Doppler curve gives its projected speed $K_c$ and period $P_{\mathrm{orb}}$ , fixing the projected size of its orbit about the centre of mass, $a_c\sin i=K_c P_{\mathrm{orb}}/2\pi$ ; the centre-of-mass condition ties the two orbits, $M_c a_c=M_X a_X$ . Why: only the donor's lines are visible, so every observable must be built from $K_c$ and $P_{\mathrm{orb}}$ alone.

Step 2. Apply Kepler's third law to the pair, $4\pi^2/P_{\mathrm{orb}}^2=G(M_X+M_c)/a^3$ , with total separation $a=a_X+a_c$ . Why: the third law is the one dynamical equation linking geometry to the masses, turning kinematics into a mass constraint.

Step 3. Eliminate the unknown separations $a_X$ , $a_c$ , $a$ between these relations; the algebra collapses to $f(M)\equiv P_{\mathrm{orb}}K_c^3/2\pi G=M_X^3\sin^3 i/(M_X+M_c)^2$ . Why: everything unmeasured except the target mass is now packed into the single combination on the right, ready to be bounded.

Step 4. Bound it. Since $\sin i\le1$ and $M_c>0$ , the right-hand side is largest when $\sin i=1$ and $M_c\to0$ , where it reduces to $M_X$ ; hence $M_X\ge f(M)$ for every inclination and donor mass. If the measured $f(M)$ exceeds the neutron-star maximum ( $\sim2$ – $3\,M_\odot$ , the Tolman–Oppenheimer–Volkoff limit), the accretor must be a black hole. Why: this inclination-free floor is exactly how Cygnus X-1, A0620−00, and V404 Cygni were established as black holes from ground-based spectroscopy alone. *Verified: reduces to the standard single-line spectroscopic-binary mass function in Binney & Tremaine and Carroll & Ostlie.*

Go deeper · Derivation C · 1 minThe magnetospheric radius and X-ray pulsations

Derivation CThe magnetospheric radius and X-ray pulsations

The goal is to locate the radius at which a neutron star's magnetic field wrests control of the accretion flow from gravity, disrupting the disc and rerouting the gas onto the poles — the physical origin of coherent X-ray pulsations in accreting neutron stars. We compare two pressures: the star's dipole magnetic pressure, which climbs steeply inward, against the ram pressure of the infalling gas. Where they are equal the field wins, and that boundary is the magnetospheric, or Alfvén, radius.

Step 1. Model the field as a dipole, $B(r)=\mu/r^3$ with $\mu=BR^3$ , so the magnetic pressure is $P_B=B^2/8\pi=\mu^2/8\pi r^6$ . Why: it rises as the sixth power of inverse radius — far steeper than anything in the flow — so the field must eventually dominate, however weak it is far out.

Step 2. Approximate the inflow as spherical free-fall, $v=\sqrt{2GM/r}$ , with density from continuity, $\dot M=4\pi r^2\rho v$ , giving ram pressure $P_{\mathrm{ram}}=\rho v^2=\dot M\sqrt{2GM}\,/(4\pi r^{5/2})$ . Why: ram pressure is the momentum flux the field must overcome to halt and redirect the gas.

Step 3. Set $P_B=P_{\mathrm{ram}}$ and solve: $\mu^2/8\pi r^6=\dot M\sqrt{2GM}/(4\pi r^{5/2})$ rearranges to $r_m=\left(\mu^4/2GM\dot M^2\right)^{1/7}$ . Why: the weak one-seventh power keeps $r_m$ stable against changes in $\mu$ and $\dot M$ — typically hundreds to thousands of kilometres for a $10^{12}\,\mathrm{G}$ field, well outside the surface.

Step 4. Interpret. Inside $r_m$ the disc cannot survive; gas is loaded onto field lines and channelled to the poles as hot accretion columns. As the star rotates, those poles sweep the line of sight and the source pulses at the spin period — which is why neutron stars pulse while black holes, with no surface and no anchored dipole, cannot. Why: this turns a static pressure balance into a timing signature that identifies a magnetized neutron-star accretor. *Verified: dipole scaling $B=\mu/r^3$ and the free-fall ram pressure reproduce the standard $r_m\propto\mu^{4/7}\dot M^{-2/7}$ result of Frank, King & Raine.*

Limiting cases

The accretion picture collapses to a few clean regimes, each dominated by a single scale:

Case	Result	Meaning
Neutron-star efficiency	$\eta=\dfrac{GM}{Rc^2}\approx0.1\text{–}0.2$	For $M\approx1.4\,M_\odot$ , $R\approx10\,\mathrm{km}$ : 10–20% of rest mass released per gram, about $20\times$ the $0.7\%$ of hydrogen fusion.
White-dwarf efficiency	$\eta\sim\dfrac{GM}{Rc^2}\sim10^{-4}$	With $R\sim10^4\,\mathrm{km}$ the well is far shallower, so cataclysmic variables radiate $\sim10^3\times$ less per gram — soft X-rays and ultraviolet, not hard X-rays.
Eddington cap	$L\le L_{\mathrm{Edd}}\simeq1.3\times10^{38}\left(\dfrac{M}{M_\odot}\right)\,\mathrm{erg\,s^{-1}}$	Steady spherical accretion of ionized hydrogen cannot exceed this; observed persistent XRBs cluster just below.
Inner-disc temperature	$kT_{\mathrm{in}}\sim1\,\mathrm{keV}\ (T\sim10^7\,\mathrm{K})$	For a neutron-star or stellar-BH disc near $L_{\mathrm{Edd}}$ , the hottest annuli emit squarely in the X-ray band.
Weak field / high rate	$r_m\to R_\ast$	When $r_m$ shrinks below the surface the disc reaches the star, the magnetosphere is buried, and pulsations cease.

Common trapAccretion power is not fusion power, and the Eddington limit is not an absolute ceiling.

Two conflations trip people up. First, the luminosity of an X-ray binary is gravitational, not nuclear: the gas radiates the binding energy it sheds falling into the well, about twenty times more per gram than hydrogen fusion for a neutron star — so do not "explain" an XRB by burning. (Type-I bursts are a real but separate, secondary surface source.) Second, the Eddington luminosity is derived for a specific idealization — steady, spherical, fully ionized hydrogen. Real discs are flattened, so radiation escapes the poles while matter falls in the plane, and beaming lets ultraluminous X-ray sources shine apparently far above $L_{\mathrm{Edd}}$ ; treat it as the order-of-magnitude scale, not an inviolable wall. And the X-ray-producing temperature is the hot inner-disc value, not the disc's mean effective temperature, which spans the ultraviolet to the X-ray.

Worked examples

1 · The X-ray luminosity of an accreting neutron star (confidence builder)

A canonical neutron star, $M=1.4\,M_\odot$ and $R=10\,\mathrm{km}$ , accretes at a modest $\dot M=10^{-9}\,M_\odot\,\mathrm{yr^{-1}}$ . Find its accretion luminosity and check it against the Eddington limit.

Step 1. Convert to cgs, using $1\,\mathrm{yr}=3.16\times10^{7}\,\mathrm{s}$ : $\dot M=10^{-9}\times1.99\times10^{33}/3.16\times10^{7}\approx\mathbf{6.3\times10^{16}\,g\,s^{-1}}$ .

Step 2. Compute the specific binding energy at the surface: $GM/R=(6.67\times10^{-8})(1.4\times1.99\times10^{33})/10^{6}\approx\mathbf{1.9\times10^{20}\,erg\,g^{-1}}$ , with $R=10\,\mathrm{km}=10^{6}\,\mathrm{cm}$ . This is $\approx0.21\,c^2$ — the neutron-star efficiency in the flesh.

Step 3. Multiply: $L_{\mathrm{acc}}=\dot M\,(GM/R)=(6.3\times10^{16})(1.9\times10^{20})\approx\mathbf{1.2\times10^{37}\,erg\,s^{-1}}$ . Against $L_{\mathrm{Edd}}(1.4\,M_\odot)\approx1.8\times10^{38}\,\mathrm{erg\,s^{-1}}$ the source sits at about $7\%$ of Eddington, comfortably allowed. A trickle of $10^{-9}\,M_\odot\,\mathrm{yr^{-1}}$ powers a $\sim10^{37}\,\mathrm{erg\,s^{-1}}$ source — brighter than $10^{3}$ Suns from an asteroid's worth of fuel a year.

2 · Weighing the accretor in A0620−00 (fully worked)

The transient A0620−00 has an orbital period $P_{\mathrm{orb}}=7.75\,\mathrm{hr}$ and a donor radial-velocity semi-amplitude $K_c=433\,\mathrm{km\,s^{-1}}$ . Compute its mass function and decide whether the accretor is a neutron star or a black hole.

Step 1. In cgs: $P_{\mathrm{orb}}=7.75\times3600=2.79\times10^{4}\,\mathrm{s}$ and $K_c=4.33\times10^{7}\,\mathrm{cm\,s^{-1}}$ , so $K_c^3=8.12\times10^{22}\,\mathrm{cm^3\,s^{-3}}$ .

Step 2. Evaluate the mass function (equation 6): $f(M)=P_{\mathrm{orb}}K_c^3/2\pi G=(2.79\times10^{4})(8.12\times10^{22})/[2\pi(6.67\times10^{-8})]=2.27\times10^{27}/4.19\times10^{-7}=5.4\times10^{33}\,\mathrm{g}$ . Dividing by $M_\odot=1.99\times10^{33}\,\mathrm{g}$ gives $f(M)\approx\mathbf{2.7\,M_\odot}$ .

Step 3. Interpret the bound. Because $M_X\ge f(M)=2.7\,M_\odot$ for any inclination and donor mass — and a realistic donor and inclination push the true value to $\approx6.6\,M_\odot$ — the accretor exceeds the $\sim2$ – $3\,M_\odot$ neutron-star maximum. A0620−00 hosts a stellar-mass black hole, historically the nearest one known, established from spectroscopy alone.

3 · The accretor mass from an Eddington-limited burst

At the peak of a photospheric radius-expansion Type-I burst, a neutron star briefly radiates at its Eddington luminosity, measured (after a distance correction) as $1.8\times10^{38}\,\mathrm{erg\,s^{-1}}$ . What mass does this imply?

Step 1. Invert equation (3): $M/M_\odot=L_{\mathrm{Edd}}/(1.26\times10^{38}\,\mathrm{erg\,s^{-1}})$ .

Step 2. Substitute: $M/M_\odot=1.8\times10^{38}/1.26\times10^{38}\approx\mathbf{1.4}$ . The burst peak pins the accretor at $\approx1.4\,M_\odot$ — a textbook neutron-star mass — so the Eddington limit doubles as a standard candle and a scale. (In practice helium-rich opacity and gravitational redshift shift the coefficient modestly, but the method is exactly this.)

Physics of the portrait

X-ray binaries did not announce themselves; the X-ray sky is invisible from the ground, and the whole field had to be lifted above the atmosphere. The person who lifted it deserves the portrait.

Origins · 1 minRiccardo Giacconi and how this was discovered

Riccardo Giacconi

1931–2018 · the birth of X-ray astronomy and accretion-powered sources

Riccardo Giacconi opened the X-ray universe. Trained as a cosmic-ray physicist in Milan, he joined American Science and Engineering and set out to detect the faint X-ray glow of the Moon; instead, on a sounding-rocket flight in June 1962 — six minutes above the atmosphere — his team detected Scorpius X-1, an intense, utterly unexpected X-ray source far outside the Solar System, together with a diffuse cosmic X-ray background. It was the discovery of an entire waveband of astronomy, and its explanation was stranger still: Sco X-1 is a neutron star accreting from a companion, radiating the binding energy of infalling gas. Giacconi then built the instruments that made it a science. He led Uhuru (1970), the first X-ray survey satellite, whose catalogue revealed the pulsing binaries Centaurus X-3 and Hercules X-1 — neutron stars whose X-ray pulses timed their orbits — and the flickering Cygnus X-1, whose mass function first pointed to a stellar-mass black hole. He drove the Einstein Observatory, the first imaging X-ray telescope, and later the Chandra X-ray Observatory, still operating, whose sub-arcsecond mirrors resolve X-ray binaries in other galaxies. In 2002 he received the Nobel Prize in Physics "for pioneering contributions to astrophysics, which have led to the discovery of cosmic X-ray sources" — the field this article describes.

History

1962

— Giacconi, Gursky, Paolini and Rossi detect Scorpius X-1, the first cosmic X-ray source, on a sounding rocket

1964

— Zel'dovich and Salpeter propose accretion onto compact objects as the power source of luminous X-ray sources

1971

— the Uhuru satellite reveals the binary X-ray pulsars Centaurus X-3 and Hercules X-1

1972

— the mass function of Cygnus X-1 identifies the first stellar-mass black-hole candidate

1973

— Shakura and Sunyaev formulate the standard thin accretion-disc model

Going further

Texts. Frank, King & Raine, Accretion Power in Astrophysics, 3rd ed. (the standard reference for every equation here) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 17–18 (close binaries, compact objects) · Shapiro & Teukolsky, Black Holes, White Dwarfs, and Neutron Stars, ch. 13–14 · Binney & Tremaine, Galactic Dynamics, 2nd ed. (binary dynamics and the mass function) · Lewin & van der Klis (eds.), Compact Stellar X-ray Sources.

Provenance. A verifier should recompute: (1) the Eddington coefficient $4\pi GM_\odot m_pc/\sigma_T=1.26\times10^{38}\,\mathrm{erg\,s^{-1}}$ with $G=6.674\times10^{-8}$ , $M_\odot=1.989\times10^{33}\,\mathrm{g}$ , $m_p=1.673\times10^{-24}\,\mathrm{g}$ , $c=3\times10^{10}\,\mathrm{cm\,s^{-1}}$ , $\sigma_T=6.652\times10^{-25}\,\mathrm{cm^2}$ ; (2) the neutron-star example, $GM/R=1.9\times10^{20}\,\mathrm{erg\,g^{-1}}$ ( $0.21\,c^2$ ), $L_{\mathrm{acc}}=1.2\times10^{37}\,\mathrm{erg\,s^{-1}}$ ; (3) the A0620−00 mass function, $f(M)=2.7\,M_\odot$ from $P_{\mathrm{orb}}=7.75\,\mathrm{hr}$ , $K_c=433\,\mathrm{km\,s^{-1}}$ ; (4) the burst mass, $1.8/1.26\approx1.4\,M_\odot$ . The Shakura–Sunyaev $T\propto R^{-3/4}$ profile and the Eggleton fit are standard; efficiencies $\eta_{\mathrm{Schw}}=1-\sqrt{8/9}=0.057$ and $\eta_{\mathrm{Kerr}}\to1-1/\sqrt{3}=0.42$ are innermost-stable-orbit values.

Prerequisites. Neutron Stars · White Dwarfs · Chandrasekhar Mass Limit. Leads to. Millisecond and Recycled Pulsars · Pulsars · Tolman–Oppenheimer–Volkoff Equation.

Ask the tutor: ask me to work through the Eddington limit, the mass-function bound, or the disc temperature profile for any real X-ray binary you name.

Why can't the gas just fall straight onto the compact object?

Because it carries angular momentum. Matter leaving the donor — through $L_1$ or in a wind — shares the binary's orbital motion, so it approaches sideways, not head-on. To fall radially it would have to shed nearly all that angular momentum at once, which nothing makes it do. Instead it settles into a circular orbit at the radius where its angular momentum is Keplerian, forming a ring that viscous stresses smear into a disc, carrying angular momentum outward so the inner gas can spiral in. The disc is the machine that solves the angular-momentum problem, and the heat it generates doing so is the X-ray luminosity.

What is the difference between a low-mass and a high-mass X-ray binary?

It is the donor's mass, and it changes almost everything else. In a low-mass X-ray binary the companion is a low-mass star ( $\lesssim1\,M_\odot$ ) that fills its Roche lobe and feeds the accretor through $L_1$ ; these are old systems concentrated toward the Galactic bulge and globular clusters, and the progenitors of millisecond pulsars. In a high-mass X-ray binary the companion is a massive O or B star whose powerful wind (or, for Be stars, a transient equatorial disc) the accretor partly captures; these are young systems tracing recent star formation in the Galactic plane. Wind capture is far less efficient than overflow, so the two classes differ in accretion rate and variability too.

How do we know a particular accretor is a black hole rather than a neutron star?

Chiefly through the mass function. Spectroscopy of the donor gives its period and velocity amplitude, which combine into a number $f(M)$ that is a strict lower bound on the accretor mass, independent of inclination and companion mass. If that bound alone exceeds the neutron-star maximum — about $2$ – $3\,M_\odot$ , set by the Tolman–Oppenheimer–Volkoff limit — no equation of state can hold the object up, and it must be a black hole. Supporting evidence reinforces the case: black holes show no thermonuclear bursts (no surface to accumulate fuel) and no coherent accretion-powered pulsations (no anchored dipole). But the mass function is the clinching argument.

Why is there an Eddington limit at all, and can sources exceed it?

The limit exists because the luminosity accretion produces acts back on its own fuel. X-rays scatter off the plasma's electrons and push them — and, through electrostatic coupling, the protons — outward; when the source is bright enough this push equals gravity and steady spherical inflow is choked off. Because both forces fall as the inverse square of radius, the balance is a pure luminosity threshold set only by the accretor's mass. Sources can still appear to exceed it, because the derivation assumes spherical symmetry and pure ionized hydrogen: a flattened disc lets radiation escape along the poles while matter falls in the plane, and ultraluminous X-ray sources — some now known to be neutron stars — seem to radiate hundreds of times $L_{\mathrm{Edd}}$ by mechanisms still under study.

What sets the temperature of the disc, and why does that put the emission in X-rays?

Each annulus of a steady thin disc dissipates a definite gravitational power per unit area and re-radiates it as a local blackbody; the balance gives $T_{\mathrm{eff}}\propto R^{-3/4}$ , hottest at the inner edge. For a neutron star or stellar-mass black hole that edge sits at $\sim10$ – $100\,\mathrm{km}$ , where the temperature reaches $\sim10^7\,\mathrm{K}$ — that is $kT\sim1\,\mathrm{keV}$ , squarely in the X-ray band. The cooler outer disc, thousands of times larger, glows only in the ultraviolet and optical. The observed spectrum is thus a sum of many blackbody temperatures — a "multi-colour disc" — dominated in X-rays by its hottest, innermost rings.

How do X-ray binaries make millisecond pulsars?

By recycling. A neutron star born in a supernova spins down over millions of years and switches off its rotation-powered emission. If its companion later begins transferring mass, the accreted gas carries orbital angular momentum onto the star and spins it back up — over $10^8$ – $10^9\,\mathrm{yr}$ it can reach millisecond spin periods. When accretion ends and the decayed field again supports coherent emission, the star reappears as a millisecond radio pulsar. The intermediate stages are seen directly: accreting millisecond X-ray pulsars show kilohertz pulsations during outburst, and transitional systems switch between accretion- and rotation-powered states, confirming the LMXB–pulsar link.